[DDS] Fix server_connection_vm_service_test timeouts

In `server_connection_common.dart`, the `server removes clients that
disconnect from the API` test spawned its own Chrome instance using
`package:devtools_shared`'s `Chrome` class without isolated profiles or
essential headless flags.

This caused the test to hang or fail flakily in container environments
(like LUCI bots) and local environments: 1. Without
`--use-mock-keychain`, headless Chrome on macOS blocks on system
credential dialogs. 2. Without `--no-sandbox`, Chrome renderer processes
can crash in restricted container environments. 3. Without
`--user-data-dir`, Chrome uses the default system profile, which can
cause it to attach to an existing open Chrome instance instead of
starting a new one, meaning the process exits immediately and the test
cannot terminate it.

Fixed by directly using `package:browser_launcher`'s `Chrome` class in
the test and passing:
* `--user-data-dir` pointing to a unique temporary directory.
* `--no-first-run` and `--no-default-browser-check` to bypass welcome prompts.
* `--no-sandbox` and `--use-mock-keychain` where appropriate.

Also wrapped the test in `try-finally` to guarantee cleanup of the
temporary profile directory.

Change-Id: I6fbe5a280524b57c635ab11ef54fa07dba2794cf
Reviewed-on: https://dart-review.googlesource.com/c/sdk/+/507600
Reviewed-by: Alexander Aprelev <aam@google.com>
Auto-Submit: Ben Konyi <bkonyi@google.com>
Commit-Queue: Ben Konyi <bkonyi@google.com>